P.1 Real Numbers and Algebraic Expressions

A ______is a collection of objects whose contents can be clearly determined. The objects in a set are called ______of the set. They are written in ______{ }.

Real Numbers

Rational Numbers Irrational Numbers

Integers

Whole Numbers

Natural Numbers

The set of ______is formed by combining the rational numbers and irrational numbers. The ______is a graph used to represent the set of real numbers.

Definition of Absolute Value

Example 1: Rewrite each expression without absolute value bars:

a) b) c) if x < 0 d)

Properties of Absolute Value

For all real numbers a and b,

1) 2) 3) 4)

5) b0 6) (called the triangle inequality)

Distance between Two Points on the Real Number Line

If a and b are any two points on a real number line, then the distance between a and b is given by or

Example 2: Find the distance between -5 and 3 on the real number line.

Algebra uses letters, such as x and y, to represent real numbers called ______. A combination of variables and numbers using the operations of addition, subtraction, multiplication, and division, as well as powers or roots, is called an ______.

Order of Operations (in order from left to right)

1)  Parentheses

2)  Exponents (powers)

3)  Multiplication and division

4)  Addition and subtraction

Example 3: Evaluate the following when x = 9

Name / Meaning / Examples
Commutative property of addition/multiplication / a + b = b + a
ab = ba
Associative property of addition/multiplication / (a+b) + c = a + (b+c)
(ab)c = a(bc)
Distributive Property of multiplication over addition / a(b + c) = ab + ac
Identity property of addition/multiplication / a + 0 = a ; 0 + a = a
a(1) = a ; 1(a) = a
Inverse property of addition/multiplication / a + (-a) = 0; (-a) + a = 0
; a0

We call the ______or ______of b. The quotient of a and b, , can be written in the form where a is the ______and b is the ______of the fraction.

Simplifying Algebraic Expressions

5x – 6y + 2

Example 4: Simplify 6(2x – 4y) + 10(4x + 3y)

Properties of Negatives:

1) (-1)a = -a 2) -(-a) = a 3) (-a)b = -ab 4) a(-b) = -ab

5) -(a + b) = -a – b 6) -(a – b) = -a + b